High-dimensional Poisson reduced-rank models have been considered for statistical inference on low-dimensional locations of the individuals based on the observations of high-dimensional count vectors. In this study, we assume sparsity on a so-called loading matrix to enhance its interpretability. The sparsity assumption leads to the use of \(L_1\) penalty, for the estimation of the loading. We provide novel computational and theoretical analyses for the corresponding penalized Poisson maximum likelihood estimation. We establish theoretical convergence rates for the parameters under weak-dependence conditions; this implies consistency even in large-dimensional problems. To implement the proposed method involving several computational issues, including nonconvex log-likelihoods, \(L_1\) penalty, and orthogonal constraints, we developed an iterative algorithm. Further, we propose a Bayesian-Information-Criteria-based penalty parameter selection, which works well in the implementation. Some numerical evidence is provided by conducting real-data-based simulation analyses and the proposed method is illustrated with the analysis of German party manifesto data.

基于高维计数向量的观察，已经考虑使用高维Poisson降秩模型对个人的低维位置进行统计推断。在这项研究中，我们假设稀疏性在所谓的加载矩阵上以增强其可解释性。稀疏性假设导致使用\（L_1 \）罚金来估计负载。我们为相应的惩罚泊松最大似然估计提供了新颖的计算和理论分析。我们建立了弱依赖条件下参数的理论收敛速度。即使在大尺寸问题中，这也意味着一致性。为了实现所提出的方法，该方法涉及多个计算问题，包括非凸对数似然\（L_1 \）惩罚和正交约束，我们开发了一种迭代算法。此外，我们提出了一种基于贝叶斯信息准则的惩罚参数选择方法，该方法在实施中效果很好。通过基于真实数据的模拟分析提供了一些数值证据，并通过对德国政党宣言数据的分析说明了所提出的方法。